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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Crank–Nicolson ： ウィキペディア英語版 Crank–Nicolson method In numerical analysis, the Crank–Nicolson method is a finite difference method used for numerically solving the heat equation and similar partial differential equations. It is a second-order method in time, it is implicit in time and can be written as an implicit Runge–Kutta method, and it is numerically stable. The method was developed by John Crank and Phyllis Nicolson in the mid 20th century.〔.〕 For diffusion equations (and many other equations), it can be shown the Crank–Nicolson method is unconditionally stable.〔. Example 3.3.2 shows that Crank–Nicolson is unconditionally stable when applied to $u\_t=au\_$.〕 However, the approximate solutions can still contain (decaying) spurious oscillations if the ratio of time step Δ times the thermal diffusivity to the square of space step, Δ, is large (typically larger than 1/2 per Von Neumann stability analysis). For this reason, whenever large time steps or high spatial resolution is necessary, the less accurate backward Euler method is often used, which is both stable and immune to oscillations. ==The method== The Crank–Nicolson method is based on the trapezoidal rule, giving second-order convergence in time. For example, in one dimension, if the partial differential equation is :$\backslash frac\; =\; F\backslash left(u,\backslash ,\; x,\backslash ,\; t,\backslash ,\; \backslash frac,\backslash ,\; \backslash frac\backslash right)$ then, letting $u(i\; \backslash Delta\; x,\backslash ,\; n\; \backslash Delta\; t)\; =\; u\_^\backslash ,$, the equation for Crank–Nicolson method is a combination of the forward Euler method at $n$ and the backward Euler method at ''n'' + 1 (note, however, that the method itself is ''not'' simply the average of those two methods, as the equation has an implicit dependence on the solution): :$\backslash frac\; -\; u\_^\}\; =$ F_^\left(u,\, x,\, t,\, \frac,\, \frac\right) \qquad \mbox :$\backslash frac\; -\; u\_^\}\; =$ F_^\left(u,\, x,\, t,\, \frac,\, \frac\right) \qquad \mbox :$\backslash frac\; -\; u\_^\}\; =$ \frac\left(discretization will also be nonlinear so that advancing in time will involve the solution of a system of nonlinear algebraic equations, though linearizations are possible. In many problems, especially linear diffusion, the algebraic problem is tridiagonal and may be efficiently solved with the tridiagonal matrix algorithm, which gives a fast $\backslash mathcal(n)$ direct solution as opposed to the usual $\backslash mathcal(n^3)$ for a full matrix. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Crank–Nicolson method」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.